
I don't really mind the incest but the prostitute (sex friends?) was kind of... (︶︿︶)

remember, he apparently told their mother that he'll take care of his brother since he's the older brother. So far within the story-telling it's been said that their mother "dropped them off" in that run down apartment to live with her lover, while she's working as a hostess.
Well living expenses don't fall from trees, school fees and clothing don't jump you from dark places, I'd think Yuuki found a way to hold his part of the deal with their mother, but assumed himself to be a bad influence on his younger brother; so he's caring for him like a goldfish, feed but don't touch.
The song Sangwoo listens to is https://youtu.be/kgl-VRdXr7I according to Koogi